Optionals IAS Mains Mathematics: Questions 204 - 210 of 283

The Equation of the central Conicoid be

… eq. (1)

Let the equation of one of the three mutually tangent planes to the central Conicoid (1) be

Where being the actual direction – cosines and

i.e.. … eq. (2)

Similarly let the equation of the other two tangent planes be

… eq. (3)

… eq. (4)

Squaring and adding, are get

Which in the required locus

Given differential equation is

It can be written as

… eq. (1)

Compare (1) with we have

Therefore, equation gives

Let

Therefore, equation become

Integrating we get, … eq. (2)

Let

And

Putting values of r and t in (1) , we get

Which is the required canonical form of given equation.

Given, Partial Differential Equation is

Where are arbitrary functions

The required solution is

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NOTE: - For convenience, we denote an n-cycle by then all the possible disjoint cycle structures of element of according to longest cycle lengths left to right, we have

Now, we know that order of a permutation of finite set written in disjoint cycle form is the least common multiple of the lengths of the cycles.

The possible order of the elements …

• Here

  T. P is non-abelian group

• The composition table is given below.

  Table Shows the Element in Composition

  0

• (i) Closure Property:

  Since all the element in composition table are elements of G, so G is closed under multiplication.

• (ii) Associativity:

  Since the transformation composition is always associative

  So G composition is also associative in G also.

• …

• We have defined by

• We know that standard basis of is

  Now

• We know that range space of T is spanned by , therefore to find a basis of range space of T, we find the L. I. vectors from
• Consider a matrix A whose rows are generators of range of T and reduce it to echelon from

  is a basis of

• Now, let

  … eq. (1)

• Now, to find the basis of null space of T is e…

• Let be any point on the surface

  … eq. (1)

  Let

• Where is the distance of from

  Let

• Where is Lagrange multiplier
• For extreme points,

  Or … eq. (2)

  And … eq. (3)

• Since x and y are both non-zero

• Take

• Take

• Hence maximum value of is and minimum value is .